Anisotropy is ubiquitously observed in many oil and gas exploration areas (e.g., the Gulf of Mexico, the North Sea, and offshore West Africa) because of preferred ordering of minerals and defects related to stresses. In these regions, often the rock properties can be characterized as transversely isotropic (“TI”) media with either a vertical or tilted axis of symmetry. Wave propagation in anisotropic media exhibits different kinematics and dynamics from that in isotropic media, thus, it requires anisotropic modeling and migration methods to image reservoirs properly for oil and gas exploration.
Three-dimensional (“3D”) anisotropic seismic modeling and migration, however, are computationally intensive tasks. Compared to prior art solutions of full elasticity, modeling and migration based on dispersion relations are computationally efficient alternatives. In one prior art method, Alkhalifah (2000), a pseudo-acoustic approximation for vertical transversely isotropic (“VTI”) media was introduced. In the approximation of that prior art method, the phase velocity of shear waves is set to zero along the vertical axis of symmetry. This simplification doesn't eliminate shear waves in other directions as described by Grechka et al. (2004). Based on Alkhalifah's approximation, several space- and time-domain pseudo-acoustic partial differential equations (PDEs) have been proposed (Alkhalifah, 2000; Zhou et al., 2006; and Du et al., 2008) for seismic modeling and migration in VTI media. These systems of PDEs are close approximations in kinematics to the solutions of full elasticity involving vector fields.
As an extension from VTI media, the axis of symmetry of a TI medium can be tilted (“TTI”) as observed in regions associated with anticlinal structures and/or thrust sheets. Zhou et al. (2006) extended their VTI pseudo-acoustic equations to a system for 2D TTI media by applying a rotation about the axis of symmetry. Consequently the phase velocity of quasi-SV waves is zero in the direction parallel or perpendicular to the tilted axis. Lesage et al. (2008) further extended Zhou's TTI system from 2D to 3D based on the same phase velocity approximation. However, these prior art pseudo-acoustic modeling and migration methods can become numerically unstable due to rapid lateral variations in tilt and/or certain rock properties (when the vertical velocity is greater than the horizontal velocity) and result in unstable wave propagation.
As one skilled in the art will appreciate, the plane-wave polarization vector in isotropic media is either parallel (for P-waves) or orthogonal (for S-waves) to the slowness vector. Except for specific propagation directions, there are no pure longitudinal and shear waves in anisotropic media. For that reason, in anisotropic wave theory the fast mode is often referred to as the “quasi-P” wave and the slow modes “quasi-S1” and “quasi-S2”.